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Anti-plane line force in nonlocal elasticity
MECHANICS RESEARCH COMMUNICATIONS Vo1.16(5), 307-309, 1989. Printed 0093-6413/89
$3.00 + .00
in the USA.
Copyright (c) 1989 Pergamon Press pie
ANTI-PLANE LINE FORCE IN NONLOCAL ELASTICITY
W a n g Rui D e p a r t m e n t of Materials Science and Engineering , The University of Science and Technology Beijing, Beijing, 100083, P. R. China. (Received 14 March 1989; accepted for print 25 May 1989)
Introduction
The point force method is a powerful method in the continuum theories of solid defects. A classical line force is the two-dimensional product of a c o n t i n u o u s u n i f o r m distribution of a point force acting along a line[l]. The present work generalizes the classical anti-plane line forcee problem to the case ofnonlocal elasticity. The analytical expressions of both, the anti-plane line force and its stress field, have been obtained by m e a n s of direct integration. None of the classical singularities exist in the stresses because of the nonlocal effect. The nonlocal line force has a Gaussian distribution quite different from the Dirac delta distribution of the classical line force and it satisfies the generalized definition. It is suggested that in the description of concentrated forces instead of using the classical Dirae delta function model , the n o n s i n g u l a r rapid attenuation function model of noniocal elasticity is adopted.
Analysis
The nonlocal stress field of l i n e a r , isotropic, nonlocal elastic solids can be expressed as [2]
tkl(x)= f a(lx'-xl)o~t(x')dvix')
(1)
0
where o kt (x')= ~,e,.,(x')Ski+ 2pek/(x')
(2)
with X and v being Lam6's constants and %t(x') being the strain tensor is the classical Hooke's law for the linear isotropic elastic solids and a( Ix' -x I) is a function of Ix' - x I. According to Eqs. (1) and (2) the nonlocal stress at a point x depends on the strain at all points x' of the body. An appropriate form of function a(Ix' -xl), suggested by Eringen 307
308
W.
RUI
[2] , i s
a(ix,_xl)=n_~( k_)3expl_k2N,_xl2/a21
(3)
(1
w h e r e k is a c o n s t a n t and a is an internal characteristic l e n g t h (e.g. atomic distance).
fz a l o n g the z - a x i s ,
For a classical a n t i - p l a n e line force
m e a s u r e d per u n i t l e n g t h and
u s i n g the cylindrical variables (r , 0 , z) , the classical (local) stress field can be expressed as 2 H r ' COS0',
Oxz ( r ' , O' ) = -
o
(r',O')
L
= -
)z
(4)
sinO',
2ltr'
S u b s t i t u t i n g (4) and (3) into (1) and u s i n g IX'-- X I = I r '2 + r 2 -- 2 rr' c o s ( O ' - - 0 ) + ( z ' - - z) 2 }1/2
we o b t a i n t = - - - I[zt xz
-5/2
2
()f -k a
3
~ a /
2rr' cos(O'-O)
t),z= - ~ n
I
a
2(z'-z)2ldz'
exp[_~o
l
} • lcos(O'-O)cosO
exp[ -
_,:o
()
I J 2,,
~ exp -
o
o
-sin(O'-O)sinO
I I
a
0
2lr'2+r2~ a l
exp
Id? dO'.
--
2[r'2+r2-
0
a /
+cos(O'-O)
sinO ] d r ' d O ' .
] 2r? cos(O'-O)
l I " [ sin(O'-O)cosO
(5)
After i n t e g r a t i n g Eqs. (5) , we obtain the nonlocal stress field of the a n t i - p l a n e line force
fz ,
fz t
XZ
--
2Hr
-k2r2/a 2 (l-e
_~2 2 / 2
fz t
yz
--
2nr
)cosO,
(l-e
(6) )sinO.
It can be seen t h a t the nonlocal stress field is not s i n g u l a r at r = 0 as long as a ; ~ 0 , and for a = 0 the solution (6) reverts to the classical solutions (4). On the basis of Eq. (17) in [3], the nonlocal anti-plane line force is F = I Q(Ix'-xl)fS(x')8(y')dv(x')
v ' w h e r e 8 is Dirac delta function. So the nonlocal anti-plane line force becomes
-k2r21a2
/(k)2 FZ =
ll
-a
f e
(7)
(8)
ANTIPLANE
LINE
FORCE
IN N O N L O C A L
ELASTICITY
309
This is the Gaussian function that differs from the Dirac delta function so long as a ~ 0. It can be verified that the foregoing results satisfy the equilibrium equation of nonlocal elasticity
t~ ,h +Fz=O.
(9)
Thus both the anti-plane line force source and its stress fields in nonlocal elasticity are obtaied. The strain field and displcement field of this problem are identical to their classical forms. Because of the nonlocal effect, the nonlocal line force has a Gaussian distribution quite different from the distribution of the classical line force. However, the total strength of the nonlocal anti-plane line force, per unit length, or
f ® I~Fzdxdy=fz,
(10)
is equal to that of the classical line force. The nonlocal anti-plane line force satisfies the generalized definition
f (txzdx+tyzdy)=0
(11)
C
with the integration circumference c bounding the area s. Discussion Reasonable physical explanation of Eqs. (8) and (10) is that interatomic long-range interaction or nonlocal effects lead to a smoothening of the singularity of classical concentrated force. This implies that in nonlocal elasticity, the expression of concentrated force generally possesses nonsingular form. In nonlocal elasticity, boundary conditions involving tractions are based on the stress tensor tat rather than oat. Therefore, if the prescribed boundary tractions are concentrated forces, the above discussion suggests to adopt the nonsingular function model of rapid attenuation instead of the classical Dirac delta function model. References 1. J. Dundurs and M. Hetenyi, J. Appl. Mech 32,671 (1965). 2. A.C. Eringen, Int. J. Eng. Sci. 15,177(1977). 3. I. Kov~ics and G. V6rgs, Physica 96B, 111 (1979).
$3.00 + .00
in the USA.
Copyright (c) 1989 Pergamon Press pie
ANTI-PLANE LINE FORCE IN NONLOCAL ELASTICITY
W a n g Rui D e p a r t m e n t of Materials Science and Engineering , The University of Science and Technology Beijing, Beijing, 100083, P. R. China. (Received 14 March 1989; accepted for print 25 May 1989)
Introduction
The point force method is a powerful method in the continuum theories of solid defects. A classical line force is the two-dimensional product of a c o n t i n u o u s u n i f o r m distribution of a point force acting along a line[l]. The present work generalizes the classical anti-plane line forcee problem to the case ofnonlocal elasticity. The analytical expressions of both, the anti-plane line force and its stress field, have been obtained by m e a n s of direct integration. None of the classical singularities exist in the stresses because of the nonlocal effect. The nonlocal line force has a Gaussian distribution quite different from the Dirac delta distribution of the classical line force and it satisfies the generalized definition. It is suggested that in the description of concentrated forces instead of using the classical Dirae delta function model , the n o n s i n g u l a r rapid attenuation function model of noniocal elasticity is adopted.
Analysis
The nonlocal stress field of l i n e a r , isotropic, nonlocal elastic solids can be expressed as [2]
tkl(x)= f a(lx'-xl)o~t(x')dvix')
(1)
0
where o kt (x')= ~,e,.,(x')Ski+ 2pek/(x')
(2)
with X and v being Lam6's constants and %t(x') being the strain tensor is the classical Hooke's law for the linear isotropic elastic solids and a( Ix' -x I) is a function of Ix' - x I. According to Eqs. (1) and (2) the nonlocal stress at a point x depends on the strain at all points x' of the body. An appropriate form of function a(Ix' -xl), suggested by Eringen 307
308
W.
RUI
[2] , i s
a(ix,_xl)=n_~( k_)3expl_k2N,_xl2/a21
(3)
(1
w h e r e k is a c o n s t a n t and a is an internal characteristic l e n g t h (e.g. atomic distance).
fz a l o n g the z - a x i s ,
For a classical a n t i - p l a n e line force
m e a s u r e d per u n i t l e n g t h and
u s i n g the cylindrical variables (r , 0 , z) , the classical (local) stress field can be expressed as 2 H r ' COS0',
Oxz ( r ' , O' ) = -
o
(r',O')
L
= -
)z
(4)
sinO',
2ltr'
S u b s t i t u t i n g (4) and (3) into (1) and u s i n g IX'-- X I = I r '2 + r 2 -- 2 rr' c o s ( O ' - - 0 ) + ( z ' - - z) 2 }1/2
we o b t a i n t = - - - I[zt xz
-5/2
2
()f -k a
3
~ a /
2rr' cos(O'-O)
t),z= - ~ n
I
a
2(z'-z)2ldz'
exp[_~o
l
} • lcos(O'-O)cosO
exp[ -
_,:o
()
I J 2,,
~ exp -
o
o
-sin(O'-O)sinO
I I
a
0
2lr'2+r2~ a l
exp
Id? dO'.
--
2[r'2+r2-
0
a /
+cos(O'-O)
sinO ] d r ' d O ' .
] 2r? cos(O'-O)
l I " [ sin(O'-O)cosO
(5)
After i n t e g r a t i n g Eqs. (5) , we obtain the nonlocal stress field of the a n t i - p l a n e line force
fz ,
fz t
XZ
--
2Hr
-k2r2/a 2 (l-e
_~2 2 / 2
fz t
yz
--
2nr
)cosO,
(l-e
(6) )sinO.
It can be seen t h a t the nonlocal stress field is not s i n g u l a r at r = 0 as long as a ; ~ 0 , and for a = 0 the solution (6) reverts to the classical solutions (4). On the basis of Eq. (17) in [3], the nonlocal anti-plane line force is F = I Q(Ix'-xl)fS(x')8(y')dv(x')
v ' w h e r e 8 is Dirac delta function. So the nonlocal anti-plane line force becomes
-k2r21a2
/(k)2 FZ =
ll
-a
f e
(7)
(8)
ANTIPLANE
LINE
FORCE
IN N O N L O C A L
ELASTICITY
309
This is the Gaussian function that differs from the Dirac delta function so long as a ~ 0. It can be verified that the foregoing results satisfy the equilibrium equation of nonlocal elasticity
t~ ,h +Fz=O.
(9)
Thus both the anti-plane line force source and its stress fields in nonlocal elasticity are obtaied. The strain field and displcement field of this problem are identical to their classical forms. Because of the nonlocal effect, the nonlocal line force has a Gaussian distribution quite different from the distribution of the classical line force. However, the total strength of the nonlocal anti-plane line force, per unit length, or
f ® I~Fzdxdy=fz,
(10)
is equal to that of the classical line force. The nonlocal anti-plane line force satisfies the generalized definition
f (txzdx+tyzdy)=0
(11)
C
with the integration circumference c bounding the area s. Discussion Reasonable physical explanation of Eqs. (8) and (10) is that interatomic long-range interaction or nonlocal effects lead to a smoothening of the singularity of classical concentrated force. This implies that in nonlocal elasticity, the expression of concentrated force generally possesses nonsingular form. In nonlocal elasticity, boundary conditions involving tractions are based on the stress tensor tat rather than oat. Therefore, if the prescribed boundary tractions are concentrated forces, the above discussion suggests to adopt the nonsingular function model of rapid attenuation instead of the classical Dirac delta function model. References 1. J. Dundurs and M. Hetenyi, J. Appl. Mech 32,671 (1965). 2. A.C. Eringen, Int. J. Eng. Sci. 15,177(1977). 3. I. Kov~ics and G. V6rgs, Physica 96B, 111 (1979).